Question

for any three vectors a b c prove that a-b b-c c-a are coplanar

Answer 1

we know, three vector $\vec{x},\vec{y}$ and $\vec{z}$ are said to be coplanar when $\vec{x}.(\vec{y}\times\vec{z})$

here, $\vec{x}=\vec{a}-\vec{b}$

$\vec{y}=\vec{b}-\vec{c}$

and $\vec{z}=\vec{c}-\vec{a}$

now, $(\vec{a}-\vec{b}).\{(\vec{b}-\vec{c})\times(\vec{c}-\vec{a})\}$

= $(\vec{a}-\vec{b}).\{\vec{b}\times\vec{c}-\vec{b}\times\vec{a}-\vec{c}\times\vec{c}+\vec{c}\times\vec{a}\}$

= $(\vec{a}-\vec{b}).\{\vec{b}\times\vec{c}-\vec{b}\times\vec{a}+\vec{c}\times\vec{a}\}$

= $[\vec{a}\vec{b}\vec{c}]-0-0-0-0-[\vec{a}\vec{b}\vec{c}]$

here, $[\vec{a}\vec{b}\vec{c}]$ means, $\vec{a}.(\vec{b}\times\vec{c})$

= 0

hence, $\vec{a}-\vec{b},\vec{b}-\vec{c}$ and $\vec{c}-\vec{a}$